<html><head><title>Volume of a vase</title><metahttp-equiv="Content-Type"content="text/html; charset=iso-8859-1"></head><bodybgcolor="#BEC7E7"text="#000000"><divalign="center"><p><b><fontsize="6"face="Comic Sans MS">Calculating the Volume of a Vase</font></b></p><palign="center"><fontsize="4"face="Comic Sans MS">Introduction</font></p><tablewidth="67%"border="0"><tr><tdwidth="39%"><imgsrc="WoodVase2.gif"width="206"height="107"></td><tdwidth="61%"><imgsrc="WoodVase1.gif"width="249"height="166"></td></tr></table><tablewidth="90%"border="0"><tr><td><divalign="center"><i>http://www.edfac.usyd.edu.au/staff/leadbeam/Woodturning.html</i></div></td></tr></table><palign="center"> </p><palign="left"><fontface="Comic Sans MS"size="2">You are to determine the
volume of the vase pictured here. The photograph on the right shows the construction
of the vase and that its height is 50 cm. </font></p><palign="left"><fontface="Comic Sans MS"size="2">Unlike practice problems
from textbooks, most real life problems involve "messy" coefficients.
Even if the function is easy to antidifferentiate, the arithmetic is usually
tedious. </font></p></div><divalign="left"><br></div><divalign="left"></div><ul></ul><divalign="center"><tablewidth="95%"border="0"><tr><tdwidth="27%"><divalign="left"><imgsrc="WoodVase.gif"width="171"height="201"></div></td><tdwidth="73%"><h3align="center"><fontface="Comic Sans MS"size="2">In this lesson
you will learn to...</font></h3><divalign="left"><ul><li><fontface="Comic Sans MS"size="2">Capture the coordinates of
points along the edge of the vase from its photograph </font></li><li><fontface="Comic Sans MS"size="2">Use those points and your
calculator to determine a function </font></li><li><fontface="Comic Sans MS"size="2">Set up the definite integral
for the volume of the vase (a solid of revolution) using your function
(need help? <ahref="http://members.tripod.com/chuwm2/revolution/volume.html"target="_blank">disk
method for calculating volume</a>)</font></li><li><fontface="Comic Sans MS"size="2">Evaluate the integral using
your calculator.</font></li></ul></div><divalign="left"><p><fontface="Comic Sans MS"size="2">You may already know how to perform
some of these. We have provided links to refresh your memory if needed.</font></p></div><p> </p></td></tr></table><p>__________________________________________________________</p><tablewidth="102%"border="0"><tr><tdcolspan="2"height="20"><divalign="center"><fontface="Comic Sans MS"size="4">Use the applet
to find the coordinate values for the edge of the vase</font></div></td></tr><tr><tdheight="380"><appletarchive="Image_and_Cursor.jar"code="Image_and_Cursor"width=366height=240align="middle">
Your browser does not support Java, so nothing is displayed.
<paramname="applet_width"value="366"><paramname="applet_height"value="240"><paramname="image_width"value="216"><paramname="image_height"value="90"><paramname="backdrop_filename"value="ROVASE2.gif"><paramname="display_placement"value="1"><paramname="display_sw"value="1"><paramname="margin_x"value="20"><paramname="back_color"value="190199231"></applet></td><tdheight="380"><ul><li><fontface="Comic Sans MS"size="2">Click anyplace on the image
to move the cursor to that point. </font></li><li><fontface="Comic Sans MS"size="2">Notice the readout at the upper
right gives the location of the cursor measured in pixels from the
lower left corner of the image. </font></li><li><fontface="Comic Sans MS"size="2">You can fine tune the location
of the cursor by clicking on the four arrows at the left, right, top,
and bottom of the map.</font></li><li><fontface="Comic Sans MS"size="2">Each click moves the cursor
one pixel in the direction indicated by the arrow. </font></li><li><fontface="Comic Sans MS"size="2">Click the blue button, labeled
"Mark point," to record the coordinates of a point.</font></li><li><fontface="Comic Sans MS"size="2">Click it each time that you
have moved the cursor to a new point whose coordinates you want to
record. </font></li><li><fontface="Comic Sans MS"size="2">Click the red button, labeled
"List points," to display the coordinates of the points
whose coordinates you have recorded.</font></li><li><fontface="Comic Sans MS"size="2">A new window will appear with
the information that you have recorded. </font></li><li><fontface="Comic Sans MS"size="2">If necessary, click the black
button, labeled "Clear points," to clear the list of recorded
points and begin again.</font></li></ul></td></tr></table><divalign="center"><p><fontface="Comic Sans MS">Now that you have coordinates from the applet,
</font></p></div></div><ul><li><divalign="center"><divalign="left"><fontface="Comic Sans MS"><ahref="regression.htm"target="_blank"><fontsize="2">Use
the regression features of a graphing calculator to find a function that
models the edge of the vase.</font></a></font></div></div></li><li><divalign="center"><divalign="left"><fontface="Comic Sans MS"size="2">Set up the <ahref="http://members.tripod.com/chuwm2/revolution/volume.html"target="_blank">definite
integral </a>that represents the volume of the vase.</font></div></div></li><li><divalign="center"><divalign="left"><fontface="Comic Sans MS"size="2">Solve the integral
using <ahref="integ-calc-direc.htm"target="_blank"> your calculator.</a></font></div></div></li><li><divalign="center"><divalign="left"><fontface="Comic Sans MS"size="2">Write your answer
with the appropriate units. </font></div></div></li><li><divalign="center"><divalign="left"><fontface="Comic Sans MS"size="2">STOP! Think about
the units of your function and the units of the actual vase. Remember
that the vase is 50 cm high.</font></div></div></li><li><divalign="center"><divalign="left"><fontface="Comic Sans MS"size="2">Convert your volume
answer into units appropriate to the actual vase. </font></div></div></li></ul><divalign="left"><palign="left"><fontface="Comic Sans MS"size="3"><ahref="solution.htm"target="_blank">Sample Solution</a></font></p><palign="left"><fontface="Comic Sans MS"size="3"><ahref="webresources.htm"target="_blank">Web
Resources</a></font></p><palign="left"><fontface="Comic Sans MS"><ahref="Credits.htm"target="_blank">Credits</a></font></p></div></body></html>
